Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity
Lakshmi Prasad Natarajan and Prasad Krishnan
Abstract—We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted asCn(r, m), and their duals are denoted asBn(r, m). The length of these codes isnm, wheren≥2, andris their ‘order’. When n = 2, Cn(r, m) is the RM code of order r and length 2m. The special case of these codes corresponding tonbeing an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer toBn(r, m)as theBerman codeandCn(r, m)as thedual Bermancode. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al.
(2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding.
I. INTRODUCTION
Reed-Muller (RM) codes [1], [2] form one of the important and well studied code families in coding theory, and have a rich algebraic structure. In [3], Kudekar et al. showed that RM codes achieve the capacity of the Binary Erasure Channel (BEC). Furthermore, Reeves and Pfister [4] have recently showed the exciting result that RM codes achieve the capacity of binary-input memoryless symmetric (BMS) channels.
In the present work, we identify a family of binary linear codes (along with its dual family) which includes the RM codes as a strict subclass. These codes are defined using a recursive construction that is similar to the Plotkin construction for RM codes. This family contains a code for each choice of three integer parameters:
(i) integersn≥2 andm≥1, which determine the length of the code, and
(ii) an integerr, with0≤r≤m, that determines the ‘order’
of the code.
We will denote the code with parameters n, r and m by Cn(r, m). The dual code, Cn(r, m)⊥, will be denoted by Bn(r, m). The length, dimension and minimum distance of Cn(r, m) are
nm,Pr w=0
m w
(n−1)w, nm−r
. (1)
The work of Lakshmi Prasad Natarajan was supported by SERB-DST via grant MTR/2019/001454. Prasad Krishnan acknowledges support from SERB- DST project CRG/2019/005572.
Lakshmi Prasad Natarajan is with the Department of Electrical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 284, India (email:
Prasad Krishnan is with the Signal Processing and Communications Re- search Center, International Institute of Information Technology, Hyderabad 500032, India (email: [email protected]).
The dual codeBn(r, m)has code parameters nm,Pm
w=r+1 m w
(n−1)w,2r+1
. (2)
If we substituten= 2in (1) and (2) we obtain the parameters of therth order RM code of length 2m, i.e., RM(r, m), and its dual RM(r, m)⊥ = RM(m−r−1, m). Indeed, we will see that the code C2(r, m) is identical to RM(r, m), and by dualityB2(r, m) = RM(m−r−1, m).
We study various basic properties ofCn(r, m)andBn(r, m) in this work. A sub-class of these codes, corresponding to the case n=p with p being an odd prime, was studied by Berman [5], and Blackmore and Norton [6] using a group algebra framework. To the best of our knowledge, Berman [5]
introduced and investigated the code Bp(r, m) and showed that its minimum distance2r+1 is better than cyclic codes of the same length (for large values ofm). Later, Blackmore and Norton [6] showed that the minimum distance of Cp(r, m) is pm−r and analyzed the state complexity of this code.
Blackmore and Norton refer toBp(r, m), the code originally designed by Berman in [5], as the Berman code. We will follow this precedence, and we will refer to Bn(r, m)as the Berman code with parameters n, r and m, and Cn(r, m) as thedual Berman code.
We comment briefly on the differences with RM codes when n ≥ 3. While RM codes are either self-orthogonal or dual- containing, Berman codes have complementary duals whenn is odd. Also, RM codes are known to be doubly transitive;
however, Berman codes with n ≥ 3 are not. Further, the minimum distances of Berman and dual Berman codes grow slowly with block length N compared to RM codes. For any choice of n ≥ 2 and any rate in (0,1), long Berman codes and their duals have mr ≈ (n−1)n . Now fixing n and letting m→ ∞, using (1), (2) and the factr≈m(n−1)/n, we see that the minimum distancedmin grows with the block length N as
dmin(Cn(r, m))∼Nn1, dmin(Bn(r, m))∼N
(n−1) nlog2n. In contrast, the minimum distance of RM codes (i.e., the case n= 2) grows approximately as the square root of N.
In the current work, we define the Berman and dual Berman codes using a recursive construction similar to the(uuu|uuu+vvv) Plotkin construction. We provide a patterned basis for the dual Berman codes, and use this basis to identify some automorphisms of Berman codes and their duals (Section II).
Finally, we utilize these automorphisms along with a result from Kumar et al. [7] to show that these codes are capacity achieving in the BEC under bit-MAP decoding (Section III).
2022 IEEE International Symposium on Information Theory (ISIT) | 978-1-6654-2159-1/22/$31.00 ©2022 IEEE | DOI: 10.1109/ISIT50566.2022.9834598
In the full version of this paper [8], we provide the proofs of various results that are not included in this present shorter version as well as a number of additional results including identifying natural generator matrices of these codes, efficient decoding algorithms up to half the minimum distance, a discrete Fourier transform (DFT) based approach to study a subclass of these codes, and further simulation results.
Furthermore, except double transitivity, we also show that they satisfy all the code properties used by Reeves and Pfister in [4]
to show that RM codes achieve the capacity of binary-input memoryless symmetric channels.
Notation: For any positive integer `, let J`K denote the set {0,1, . . . , `−1}. The notation 000 denotes a zero-vector or a zero-matrix of appropriate size. For two vectors aaa, bbb their concatenation is denoted by (aaa|bbb). We denote an n-length vector aaa by its components asaaa= (ai :i∈JnK). For some S ⊂JnK,we denote the vector with componentsai:i∈S as a
a
aS. Ifaaais a nm-length vector for some m≥1, we also use the concatenation representationaaa= (aaa0|aaa1|. . .|aaan−1), where a
a
al : l ∈ JnK are subvectors of length nm−1. The individual components ofaaalwould be then denoted asal,i:i∈Jnm−1K.
II. BERMANCODES AND THEIRDUALS
A. Recursive Definition of Berman & Dual Berman Codes We now proceed to give the definitions of the codes pre- sented in this work. For some positive integersn≥2 andm, for some non-negative integerr such that 0≤r≤m, define the family of codes Bn(r, m)⊆Fn2m recursively as follows.
Bn(m, m),{000}, Bn(0, m),{ccc∈Fn2m :X
i
ci= 0}.
Form≥2and1≤r≤m−1,
Bn(r, m),{(vvv0|vvv1|. . .|vvvn−1) :vvvl∈Bn(r−1, m−1)
∀l∈JnK, X
l∈JnK
v
vvl∈Bn(r, m−1)}.
We refer to the code Bn(r, m) as the Berman code with parameters n, m, andr. We similarly define the code family Cn(r, m)⊆Fn2m recursively.
Cn(m, m),Fn2m, Cn(0, m),{(0, . . . ,0),(1, . . . ,1)}.
Form≥2and1≤r≤m−1,
Cn(r, m),{(uuu+uuu0|uuu+uuu1|. . .|uuu+uuun−2|uuu) : u
u
ul∈Cn(r−1, m−1)∀l∈Jn−1K, uuu∈Cn(r, m−1)}.
We shall refer to Cn(r, m) as the dual Berman code with parameters n, m and r (this nomenclature will be validated in Theorem II.1, where we show that the codesBn(r, m)and Cn(r, m) are dual to each other). Also, observe that when n= 2, the codeC2(r, m)is then defined asC2(r, m) ={(uuu+
uu
u0|uuu) :uuu0 ∈ Cn(r−1, m−1), uuu ∈ Cn(r, m−1)}, which
coincides with RM(r, m). Thus the class of codes Cn(r, m) includes the Reed-Muller codes.
Example II.1. We give some specific examples of the codes defined above.
• By the recursive definition, the single parity check code is used as the building block along with a global parity to obtain the following code forn= 3, m= 2, r= 1,
B3(1,2) ={(vvv0|vvv1|vvv0+vvv1) :vvv0, vvv1∈B3(0,1)}
={(v00, v01, v00+v01|v10, v11, v10+v11|
v00+v10, v01+v11, v00+v01+v10+v11) : vij ∈F2,∀i, j∈ {0,1}}.
• The code C3(1,2) is shown below as per the recursive construction. It is easy to verify that it is dual toB3(1,2).
C3(1,2) ={(u00, u00, u00|u10, u10, u10|0,0,0)+
(u0, u1, u2|u0, u1, u2|u0, u1, u2) : u00, u10, u0, u1, u2∈F2}.
It is clear that the codesCn(r, m)andBn(r, m)are linear.
We shall now provide various properties of these codes. The techniques involved in the proofs (available in [8]) are similar to those for RM codes, mainly involving induction on the parameter m. The parameters of Cn(r, m) were previously derived in [6, Remark 2.4], and the parameters of Bp(r, m) for odd primepwere derived in [5, Theorem 2.2], both using a group algebra framework.
Theorem II.1. (Basic properties ofCn(r, m)and Bn(r, m).) 1) Containment property: For1≤r≤m,
a) Bn(r, m)⊂Bn(r−1, m).
b) Cn(r−1, m)⊂Cn(r, m).
2) Dimension:
dim(Bn(r, m)) =
m
X
w=r+1
m w
(n−1)w,
dim(Cn(r, m)) =
r
X
w=0
m w
(n−1)w.
3) Duality:Cn(r, m)⊥=Bn(r, m).
4) Minimum Distance:
dmin(Bn(r, m)) = 2r+1,∀0≤r≤m−1, dmin(Cn(r, m)) =nm−r,∀0≤r≤m.
B. A Patterned Basis for the Berman Code
Our next goal is to obtain a special patterned basis for Bn(r, m). Towards that end, we now give some notation to work with the indices of vectors in Fn2m. This notation will also be used in the forthcoming section. Let G = JnK = {0,1, . . . , n−1}. We then identify the nm coordinates of an arbitrary vector vvv ∈ Fn2m using the m-tuples in Gm, i.e., vvv = (viii : iii ∈ Gm). We also write vvv as a concatenation of n vectors from Fn2m−1, denoted by vvv = (vvv0|. . .|vvvn−1). The subvectorvvvl∈Fn2m−1 is then recursively identified as follows.
• For anyiii0 ∈Gm−1, the component ofvvvl indexed byiii0 is identified as vl,iii0 =v(iii0|l) which is the component of v
v
v indexed by (iii0|l)∈Gm.
We also need the following definition of a patterned-vector in Fn2m. For m ≥ 1 and some iii0 ∈ Gm, define the vector cccm(iii0)∈Fn2m as consisting of the entries
cm(iii0)iii=
(1 ifsupp(iii)⊆supp(iii0)andiiisupp(iii)=iii0supp(iii) 0 otherwise,
∀iii∈Gm, wheresupp(iii),{l∈JmK:il6= 0} is the support set. That is, the vectorcccm(iii0)has1exactly in those coordinates iii∈ Gm which satisfyil =i0l for all l :il 6= 0. We give an example to illustrate the definition of cccm(iii0).
ExampleII.2. Considerm= 3, n= 3, andG={0,1,2}. Let iii0= (1,2,0)∈G3. The components of the vectorccc3(iii0)∈F272
are as follows.
ccc3(iii0)iii=
(1 foriii∈ {(0,0,0),(1,0,0),(0,2,0),(1,2,0)}, 0 otherwise.
We are now ready to show a patterned basis for Bn(r, m).
Let wH(iii),|supp(iii)| be the Hamming weight ofiii.
Lemma II.1. Form≥1, and 0≤r≤m−1, consider the collection of elements in Fn2m given by
BBn(r, m) ={cccm(iii0) :
∀iii0∈Gm such thatr+ 1≤wH(iii0)≤m}. (3) Then the collectionBBn(r, m)is a basis for Bn(r, m).
Proof: We first show that the vectors in BBn(r, m) are linearly independent. Let B0 be any non-empty subset of vectors fromBBn(r, m). Note that each vector inB0is of the formcccm(iii0)for some uniqueiii0 ∈Gm withwH(iii0)≥r+ 1, by the construction of set BBn(r, m).
We will show that theF2-sum of the vectors fromB0cannot be zero, which suffices to show that BBn(r, m)is a linearly independent set of vectors.
Let cccm(iiid) ∈ B0 be such that wH(iiid) ≥ wH(iii0) for any cccm(iii0)∈B0. Thus,cccm(iiid)is a maximal element inB0 in this sense. Note that such a maximal elementcccm(iiid)will always exist for any non-empty B0⊆BBn(r, m).
We observe the following by the definition of the vectors cccm(iii0) ∈BBn(r, m). For anycccm(iii0) ∈B0, if cm(iii0)iiid = 1, we must havesupp(iiid)⊆supp(iii0)andiii0supp(iiid)= (iiid)supp(iiid). By the maximality ofcccm(iiid), we must havewH(iiid) =wH(iii0).
Hence, by these observations, we must have thatiii0=iiid. Thus, the sum of vectors in B0 cannot be 000 (as theiiithd coordinate in the sum cannot be 0). Thus, the vectors in BBn(r, m)are linearly independent.
Also, we see that|BBn(r, m)|=Pm w=r+1
m w
(n−1)w= dim(Bn(r, m)). Thus, showing thatBBn(r, m) ⊂ Bn(r, m) will conclude the proof. The rest of the proof is devoted to showing this statement.
Consider an arbitrary cccm(iii0) ∈ BBn(r, m). Note that wH(cccm(iii0)) = 2wH(iii0)≥2r+1, by definition. Thus,cccm(iii0)∈
Bn(0, m)has even weight. Thus the statement holds forr= 0 for anym. Thus, the statement holds form= 1.
Now we prove the statement for r ≥ 1, m ≥ 2 assuming it holds for m − 1. Recall that we can use the concatenation representation for cccm(iii0) as cccm(iii0) = (cccm(iii0)0|cccm(iii0)1|. . .|cccm(iii0)n−1). We consider two cases.
Case (a): m−1 ∈ supp(iii0). Let i0m−1 = l0 ∈ G\ {0}.
Thus, for some iii ∈ Gm, if im−1 ∈ G \ {l0,0}, then cccm(iii0)iii = 0. This means cccm(iii0)l = 000 ∈ Fn2m−1 if l /∈ {l0,0}. Further if l ∈ {l0,0}, then we can observe that cccm(iii0)l = cccm−1(iii0
Jm−1K) ∈ Fn2m−1, where we recall the notation iii0
Jm−1K = (i0l : l ∈ Jm−1K). As supp(iii0
Jm−1K) = supp(iii0)\ {m−1}, thus r ≤wH(iii0
Jm−1K)≤ m−1, which meanscccm−1(iii0
Jm−1K)∈BBn(r−1, m−1). By the induction hypothesis, we thus havecccm−1(iii0
Jm−1K)∈Bn(r−1, m−1).
Further,P
l∈JnK
cccm(iii0)l=cccm(iii0)0+cccm(iii0)l0 = 000∈Bn(r, m−
1). Thus the two conditions in the definition of Bn(r, m)are satisfied, and thuscccm(iii0)∈Bn(r, m).
Case (b): m−1 ∈/ supp(iii0). In this case, a necessary condition forcccm(iii0)iii= 1is thatm−1∈/ supp(iii). This means we havecccm(iii0)l= 000 ifl6= 0,andcccm(iii0)l=cccm−1(iii0
Jm−1K)∈ Fn2m−1 for l = 0. Now, as supp(iii0
Jm−1K) = supp(iii0), this means that r+ 1≤wH(iii0
Jm−1K) =wH(iii0)≤m−1. Hence, cccm−1(iii0
Jm−1K) ∈ BBn(r, m−1) and thus cccm−1(iii0
Jm−1K) ∈ Bn(r, m−1) by the induction hypothesis. By Theorem II.1 (part 1), this meanscccm−1(iii0
Jm−1K)∈Bn(r−1, m−1). It is thus clear that the conditions in the definition ofBn(r, m)are satisfied by the vectorcccm(iii0).This concludes the proof.
Example II.3. For the code B3(1,3), with the coordinates indexed by {0,1,2}3, Lemma II.1 shows that the following collectionBB3(1,3)consisting of 20vectors is a basis.
[
a,b,c∈{1,2}
{ccc3((a, b,0)), ccc3((0, a, b)), ccc3((a,0, b)), ccc3((a, b, c))}
C. Some useful automorphisms ofBn(r, m)and Cn(r, m)
The results in this sub-section specify some automorphisms of the code Bn(r, m)(and hence for Cn(r, m), by duality).
Lemma II.2. Let σ be any permutation of the set G. The following permutation πm−1,σ on the m-tuples in Gm is an automorphism ofBn(r, m).
πm−1,σ: (i0, . . . , im−2, im−1)7→(i0, . . . , im−2, σ(im−1)).
Proof: Letvvv = (vvv0|. . .|vvvn−1) be an arbitrary codeword in Bn(r, m). We want to show that the vector vvv0, with coordinatesviii0=vπm−1,σ(iii), also lies inBn(r, m).
To see this, observe that if we writevvv0 as (vvv00|. . .|vvv0n−1), for anyl∈JnKwe have thatvvv0l=vvvl0 for precisely that unique l0 such thatσ(l0) =l. Thus, the subvectors ofvvv0 are precisely the same as those invvv, only their positions are permuted. Thus vvv0 satisfies the two conditions in the definition of Bn(r, m).
Hencevvv0∈Bn(r, m), which completes the proof.
Lemma II.3. For anyt∈Jm−1K, the permutationβtonGm defined below is an automorphism of Bn(r, m).
βt: (i0, . . . , im−1)7→(i0, . . . , it−1, im−1, it+1, . . . , im−2, it).
Proof: Clearly, the statement is true for r = m. Hence we assume r ≤ m−1. Recalling the definition of the set BBn(r, m)in (3), to show the lemma, it is sufficient to show that for each cccm(iii0) ∈ BBn(r, m), the permuted vector ccc0 defined below is also inBBn(r, m).
ciii0 =cm(iii0)βt(iii), ∀iii∈Gm. (4) We shall in fact prove that ccc0 =cccm(βt(iii0))). The proof will then be complete asβtis a one-one map.
Firstly we observe that the following two statements are equivalent for any iii ∈ Gm, because βt is a self-inverse permutation.
• supp(βt(iii))⊆supp(iii0)andβt(iii)supp(βt(iii))=iii0supp(βt(iii)), are both true.
• supp(iii) ⊆ supp(βt(iii0)) and iiisupp(iii) = βt(iii0)supp(iii), are both true.
Now, using the above equivalence and (4), we have,
ciii0 =
(1 if supp(iii)⊆supp(βt(iii0))&iiisupp(iii)=βt(iii0)supp(iii), 0 otherwise,
(5) for alliii∈Gm. Clearly, by (5), we see thatccc0 is precisely the vector cccm(βt(iii0)). Since wH(βt(iii0)) = wH(iii0) ≥r+ 1, we have thatcccm(βt(iii0))∈BBn(r, m)as well. This completes the proof.
We now summarize the results from the above two lemmas.
We use Smto denote the symmetric group of degree m, i.e., Smis the group of all permutations onJmK.
Theorem II.2. Letσ0, . . . , σm−1 be any permutations of the set G, and let γ ∈ Sm. The following permutations on Gm are automorphisms ofBn(r, m)andCn(r, m).
(i0, . . . , im−1)→(σ0(i0), . . . , σm−1(im−1)), (i0, . . . , im−1)→(iγ(0), . . . , iγ(m−1)).
Proof: It is sufficient to show that the permutations in the statement are automorphisms of Bn(r, m)because of the duality of Bn(r, m)andCn(r, m).
Part 1): For any t∈Jm−1K, the permutation
(i0, . . . , im−1)→(i0, . . . , it−1, σt(it), it+1, . . . , im−1), is identical with the compositionβtπm−1,σtβt, whereπm−1,σt andβtare as defined in Lemma II.2 and Lemma II.3 respec- tively. Thus, the permutation
(i0, . . . , im−1)→(σ0(i0), . . . , σm−1(im−1)), (6)
is identical with the composition
πm−1,σm−1
Q
t∈Jm−1K
βtπm−1,σtβt
!
. Since the set of automorphisms ofBn(r, m)form a group under composition,
by Lemmas II.2 and II.3, we have that (6) is an automorphism of Bn(r, m).
Part 2): It is known (see, for example, [9]) that any permutation γ ∈ Sm can be generated by a composition of transpositions, where a transposition refers to a permutation which interchanges one element of JmK with another, and leaves the other elements as is. Observe thatβt as in Lemma II.3 is precisely the transposition that interchanges t with m−1. Further, a transposition that interchanges two distinct elementst1, t2∈Jm−1Kcan be obtained as the composition βt2βt1βt2. This completes the proof following Lemma II.3 and because the automorphisms ofBn(r, m) form a group.
III. CAPACITY-RELATEDPROPERTIES
A. Rate ofBn(r, m)and Cn(r, m) The rate of Cn(r, m) is Rn(r, m) ,
Pr
w=0(mw)(n−1)w
nm ,
which is equal to the fraction of vectors in Gm with weight at the mostr. Similar to RM codes [4], the rate ofCn(r, m) can be seen to be equal to the cumulative distribution function of a binomial random variable with m trials and probability of success (n−1)/n (please see [8] for more details). The Berry-Esseen inequality [10] can be used to approximate this distribution function with that of the standard Gaussian.
Let Q(x) = R∞ x
√1
2πe−t2/2dt be the tail probability of the standard Gaussian distribution. The Berry-Esseen inequality guarantees that there exists a constant κ > 0, that depends only onn, such that
Rn(r, m)− 1−Q
r−mµ√ mσ2
≤√κm, (7) whereµ= (n−1)/nandσ2= (n−1)/n2. We also note that 1−Rn(r, m) is the rate of Bn(r, m). Hence, from (7), we deduce that the rate ofBn(r, m)is Q
r−mµ√ mσ2
+O
√1 m
. B. Achieving the BEC Capacity
Kumar, Calderbank and Pfister [7, Theorem 19] use code automorphisms to provide a sufficient condition for a code to achieve the capacity of BEC under bit-MAP decoding. This condition is less demanding than requiring double transitivity, which was the property used in [3] to prove RM codes achieve BEC capacity. To use this result on a sequence of codes with increasing block lengths, we require the following
1) the rates of the sequence of codes must converge to a value in(0,1),
2) each code in this sequence must be transitive,
3) for each code the orbit of the coordinates under a sub- group of automorphisms (those automorphisms that fix an arbitrarily chosen coordinate) must be sufficiently large.
We will now apply this result to the family of codes {Cn(r, m)}. A similar result holds for{Bn(r, m)}.
1) Code Sequence with Converging Rate: For a givenn≥2 andR∗ ∈(0,1), consider a sequence of codes{Cn(rl, ml)}
withml→ ∞andrl=mlµ+Q−1(1−R∗)√
mlσ2+o(√ ml).
Using (7) we note that the rateRn(rl, ml)→R∗asml→ ∞.
Hence, for anyR∗∈(0,1)there exists a sequence ofCn(r, m) codes with increasing lengths, and rates converging toR∗.
2) Transitivity: We now use Theorem II.2 to observe that for any choice of parameters n, r, m, the code Cn(r, m) is transitive. Consider any choice of coordinatesiii, jjj∈Gm. We need to show that there is a code automorphism that mapsiii tojjj. Letσ0, . . . , σm−1be permutations of the setGsuch that
σ0(i0) =j0, . . . , σm−1(im−1) =jm−1.
Applying Theorem II.2 for this choice ofσ0, . . . , σm−1shows that Cn(r, m)is indeed transitive.
3) Orbits Under a Subgroup of Automorphisms: Let G0
be the subgroup of automorphisms ofCn(r, m)that fixes the coordinate 000 ∈ Gm. We want a lower bound on the size of the orbits ofiii∈Gm\ {000} under the action ofG0, which is
Or,m(iii),{π(iii) :π∈ G0}.
We will identify a subset ofOr,m(iii)to obtain this lower bound.
Consider any iii6= 000 and anyjjj ∈Gm such that supp(iii) = supp(jjj). There existmpermutations ofG,σ0, . . . , σm−1, such that
σl(il) =jl andσl(0) = 0 for alll∈JmK.
Using Theorem II.2, we see that the map (k0, . . . , km−1)→ (σ0(k0), . . . , σm−1(km−1)) is an automorphism of Cn(r, m) that fixes 000 and sends iiitojjj. We thus conclude that Or,m(iii) contains alljjj such thatsupp(jjj) =supp(iii).
Now, for a giveniii6= 000, consider anyjjjsuch that wH(jjj) = wH(iii). Clearly, there exists a permutation γ ∈Sm such that supp(jjj) = supp(γ(iii)). From our argument in the previous paragraph, jjj∈Or,m(γ(iii)). Sinceγ is a code automorphism that fixes000, we see that γ(iii) itself belongs to Or,m(iii), and therefore, jjj ∈ Or,m(iii). We have thus showed that for any iii6= 000,Or,m(iii)⊇ {jjj∈Gm:wH(jjj) =wH(iii)}. Hence,
|Or,m(iii)| ≥ m
wH(iii)
(n−1)wH(iii).
Note that for any n≥3, and anyiii∈Gm\ {000}, we have
|Or,m(iii)| ≥2m.
We are now ready to apply [7, Theorem 19]. Let n ≥ 3.
Consider a sequence of codes{Cn(rl, ml)}withml→ ∞and rates converging toR∗∈(0,1). All the codes in this sequence are transitive, and they satisfy
iii∈Gminml\{000}|Orl,ml(iii)| → ∞ asl→ ∞.
These are precisely the sufficient conditions identified in [7]
to guarantee that this sequence of codes has a vanishing bit erasure probability under bit-MAP decoding in the BEC for any channel erasure probability < (1−R∗). Note that a similar result holds for Berman codes {Bn(r, m)}.
IV. SIMULATIONRESULT
In our simulations we have compared codes with reasonably close rates and lengths. To account for the difference in the rates we use−(1−R), instead of, as the horizontal axis in our plots (whereRis the code rate). Note that−(1−R)is the difference between the actual channel erasure probability and
-0.04 -0.02 0
- (1-R) 10-5
10-4 10-3 10-2 10-1 100
PB
Berman Code RM Code
-0.04 -0.02 0
- (1-R) 10-5
10-4 10-3 10-2 10-1 100
Pb
Berman Code RM Code
Fig. 1. The block and bit erasure rates ofB3(5,7)andRM(4,11)in BEC.
the capacity limit (i.e.,1−R is the highest possible channel erasure probability that any code of rateR can withstand).
We compare the block erasure ratePB (under block-MAP decoding) and bit erasure rate Pb (under bit-MAP decoding) ofB3(5,7)withRM(4,11)in Fig. 1. The parameters of these two codes are [2187,576,64] and [2048,562,128], respec- tively. Note the similarity in their bit erasure rate performance for Pb≥10−5.
In [8], we compare the performances of the dual Berman codeC3(5,7)(with parameters[2187,1611,9]) andRM(6,11) (parameters [2048,1486,32]). While these two codes have similarPb, thePB curve ofC3(5,7)exhibits a high floor due to its small minimum distance.
V. DISCUSSION
We identified a family of codes that includes the RM codes [1], [2] (n = 2) and whose properties are similar to RM codes. While the similarity of Cn(r, m) and Bn(r, m) to RM codes is striking, there are some key differences as well, especially in terms of the minimum distance and the automorphism group. Our guarantees on capacity achievability in the BEC are in the sense of vanishing bit erasure probability Pb. It is not clear if these codes have vanishing block erasure probability PB under block-MAP decoding for rates close to capacity limit. It is possible that some of the differences of Cn(r, m) and Bn(r, m) with RM codes might offer ad- vantages. For large block lengths, the minimum distance of Cn(r, m)is significantly smaller than that of RM codes. This implies that its dual Bn(r, m) has a parity check matrix that is considerably sparser than the parity-check matrix of RM codes. This sparsity might be useful in designing low complexity iterative decoders forBn(r, m), see [11].
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